Deterministic walks on a square lattice Ra´ ul Rechtman Instituto de Energ´ ıas Renovables, Universidad Nacional Aut´ onoma de M´ exico, Temixco, Mor., Mexico rrs@cie.unam.mx Advances in Nonequilibrium Statistical Mechanics: large deviations and long-range correlations, extreme value statistics, anomalous transport, and long-range interactions, The Galileo Galilei Institute for Theoretical Physics, Arcetri, Florence, July 1, 2014 Ra´ ul Rechtman (IER-UNAM) Deterministic walks 1 / 53
Contents Introduction 1 A walker on an initially ordered flipping rotor landscape 2 A walker on a partially ordered flipping rotor landscape 3 Two walkers on an initially ordered flipping rotor landscape 4 A walker on an initially disordered flipping rotor landscape 5 Concluding remarks 6 Ra´ ul Rechtman (IER-UNAM) Deterministic walks 2 / 53
Introduction Deterministic walks Lorentz gas. A walker on a landscape. The walker interacts with the landscape during the walk. Landscape = 2 d square lattice with obstacles. Complex system. Simple model of anomalous transport. H. A. Lorentz, Proc. Amst. Acad. 7 438 (1905). E. G. D. Cohen, L. Bunimovich, J. P. Boon, X. P. Kong, P. M. Binder, H-F. Meng. Ra´ ul Rechtman (IER-UNAM) Deterministic walks 3 / 53
Introduction The Ehrenfest’s wind-tree model (1911) df i dt = k ( f i +1 + f i − 1 − 2 f i ) , i = 0 , . . . 3 � f 0 (0) = 1 , f 1 (0) = f 2 (0) = f 3 (0) = 0 , f i = 1 , i f 0 = 1 1 + e ( − 2 kt ) � 2 , f 1 = f 3 = 1 � � 1 + e ( − 2 kt ) � � 1 − e ( − 2 kt ) � , 4 4 f 2 = 1 1 − e ( − 2 kt ) � 2 . � 4 1 f 0 v 1 f 1 , f 3 f 2 0.75 0.5 v 2 v 0 0.25 0 v 3 t P. Ehenfest, T. Ehrenfest, Begriffliche Grundlagen der Statistische Auffassung in der Mechanik, Encyklop¨ adie der Mathematische Wissenschaften vol. 4 pt 32 (Leipzig: Teubner ), 1911. Engl. Trans. M. J. Moravcsik, The Conceptual Foundations of the Statistical Approach in Mechanics , Ithaca, Cornell University Press, 1959. R. Rechtman, A. Salcido, A. Calles, EPL 12 27 (1991). Ra´ ul Rechtman (IER-UNAM) Deterministic walks 4 / 53
Introduction a X. P. Kong, E. G. D. Cohen, Phys. Rev. B 40 , 4838 (1989). Th. Ruijgrok, E. G. D. Cohen, Phys. Lett. A, 133 415 (1988). H-F. Meng, E. G. D. Cohen, Phys. Rev. E 50 2482 (1994). Ra´ ul Rechtman (IER-UNAM) Deterministic walks 5 / 53
Introduction a X. P. Kong, E. G. D. Cohen, Phys. Rev. B 40 , 4838 (1989). Th. Ruijgrok, E. G. D. Cohen, Phys. Lett. A, 133 415 (1988). H-F. Meng, E. G. D. Cohen, Phys. Rev. E 50 2482 (1994). Ra´ ul Rechtman (IER-UNAM) Deterministic walks 5 / 53
Introduction flipping mirror landscape flipping rotor landscape right mirror left mirror right rotor left rotor Ra´ ul Rechtman (IER-UNAM) Deterministic walks 6 / 53
Introduction flipping mirror landscape flipping rotor landscape a a Ra´ ul Rechtman (IER-UNAM) Deterministic walks 7 / 53
Introduction flipping mirror landscape flipping rotor landscape a a Ra´ ul Rechtman (IER-UNAM) Deterministic walks 7 / 53
Introduction flipping mirror landscape flipping rotor landscape a a Ra´ ul Rechtman (IER-UNAM) Deterministic walks 7 / 53
Introduction flipping mirror landscape flipping rotor landscape a a Ra´ ul Rechtman (IER-UNAM) Deterministic walks 7 / 53
Introduction flipping mirror landscape flipping rotor landscape a a Ra´ ul Rechtman (IER-UNAM) Deterministic walks 7 / 53
Introduction flipping mirror landscape flipping rotor landscape a a Ra´ ul Rechtman (IER-UNAM) Deterministic walks 7 / 53
Introduction flipping mirror landscape flipping rotor landscape a a Ra´ ul Rechtman (IER-UNAM) Deterministic walks 7 / 53
Introduction flipping mirror landscape flipping rotor landscape a a Ra´ ul Rechtman (IER-UNAM) Deterministic walks 7 / 53
Introduction A walker moves on a 2 D square lattice, the landscape, in discrete time steps to a nearest neighbor site according to the landscape. In so doing, he alters the landscape locally. At time t the walker is at ( x , y ) with one of four velocities v 0 = (1 , 0), v 1 = (0 , 1), v 2 = ( − 1 , 0), or v 3 = (0 , − 1). The state of the landscape, m ( x , y ), is either 1 or -1 and after the walker passes, m changes sign. The landscape is made of flipping rotors or flipping mirrors. In the first case, the particle turns right or left according to m ( x , y ), and in the second one, the particle is reflected by a “mirror” with an inclination of 45 ◦ or 135 ◦ . flipping mirror landscape v 1 � walker reflects from a mirror at 45 ◦ 1 m ( x , y ) = walker reflects from a mirror at 135 ◦ − 1 v 2 v 0 flipping rotor landscape walker rotates 90 ◦ to the right � 1 m ( x , y ) = walker rotates 90 ◦ to the left − 1 v 3 Ra´ ul Rechtman (IER-UNAM) Deterministic walks 8 / 53
Introduction flipping mirror landscape flipping rotor landscape v ′ v ′ x = mv y x = mv y v ′ v ′ y = + mv x y = − mv x m ′ = − m m ′ = − m x ′ = x + v ′ x ′ = x + v ′ x x y ′ = y + v ′ y ′ = y + v ′ y y The primed (unprimed) quantities refer to t + 1 ( t ). Ra´ ul Rechtman (IER-UNAM) Deterministic walks 9 / 53
Introduction At t = 0, m ( x , y ) = 1 ∀ x , y and the walker is in the center of the lattice with v = v 1 . flipping mirror landscape flipping rotor landscape 80 80 35 30 25 20 40 40 15 10 5 0 0 0 0 40 80 0 40 80 The walker has moved during 9,000 time steps. The The walker moves alternatively one step vertically, one colors show the number of times each site has been horizontally. visited. Ra´ ul Rechtman (IER-UNAM) Deterministic walks 10 / 53
Contents Introduction 1 A walker on an initially ordered flipping rotor landscape 2 A walker on a partially ordered flipping rotor landscape 3 Two walkers on an initially ordered flipping rotor landscape 4 A walker on an initially disordered flipping rotor landscape 5 Concluding remarks 6 Ra´ ul Rechtman (IER-UNAM) Deterministic walks 11 / 53
Initially ordered flipping rotor landscape t = 100 80 7 6 5 4 40 3 2 1 0 0 0 40 80 Ra´ ul Rechtman (IER-UNAM) Deterministic walks 12 / 53
Initially ordered flipping rotor landscape t = 1000 80 18 15 12 40 9 6 3 0 0 0 40 80 Ra´ ul Rechtman (IER-UNAM) Deterministic walks 12 / 53
Initially ordered flipping rotor landscape t = 3000 80 25 20 15 40 10 5 0 0 0 40 80 Ra´ ul Rechtman (IER-UNAM) Deterministic walks 12 / 53
Initially ordered flipping rotor landscape t = 5000 80 30 24 18 40 12 6 0 0 0 40 80 Ra´ ul Rechtman (IER-UNAM) Deterministic walks 12 / 53
Initially ordered flipping rotor landscape t = 7000 80 30 24 18 40 12 6 0 0 0 40 80 Ra´ ul Rechtman (IER-UNAM) Deterministic walks 12 / 53
Initially ordered flipping rotor landscape t = 9000 80 30 24 18 40 12 6 0 0 0 40 80 Ra´ ul Rechtman (IER-UNAM) Deterministic walks 12 / 53
Initially ordered flipping rotor landscape t = 11000 80 30 24 18 40 12 6 0 0 0 40 80 Ra´ ul Rechtman (IER-UNAM) Deterministic walks 12 / 53
Initially ordered flipping rotor landscape At t = 0, m ( x , y ) = 1 ∀ ( x , y ) and the walker is in the center of the lattice with v = v 1 . v ′ t = 11000 x = mv y 80 v ′ y = − mv x 30 m ′ = − m 24 x ′ = x + v ′ x y ′ = y + v ′ 18 40 y 12 v ′ = v k − m m ′ = − m 6 r ′ = r + v ′ 0 0 0 40 80 After almost 10,000 time steps, T 0 , the walker begins to move periodically. Every 100 or so time steps, T 1 , it moves 2 sites horizontally and 2 vertically. Ra´ ul Rechtman (IER-UNAM) Deterministic walks 13 / 53
Initially ordered flipping rotor landscape T 0 = 9 , 977. For t > T 0 the particle moves periodically with period T 1 = 104. 75 x y 60 45 30 15 0 0 4,000 8,000 12,000 T 0 t Ra´ ul Rechtman (IER-UNAM) Deterministic walks 14 / 53
Initially ordered flipping rotor landscape For t > T 0 , the walker moves periodically with period T 1 = 104 and x and y diminish by 2 with a speed √ u = 2 2 / 104. 60 x y 45 30 15 T 0 + T 1 T 0 + 2 T 1 T 0 + 3 T 1 T 0 t The two straight lines have slope − 2 / 104. Ra´ ul Rechtman (IER-UNAM) Deterministic walks 15 / 53
Initially ordered flipping rotor landscape t = T 0 t = T 0 + T 1 51 49 49 47 47 45 45 43 43 41 41 39 20 22 24 26 18 20 22 24 At t = T 0 the walker is at the site marked by the red circle (left Fig.) with v = v 2 . At t = T 0 + T 1 the walker is at the site marked by the red circle (right Fig.) with v = v 2 . The walker moved two sites to the left and two down. In doing so the walker prepared the landscape in such a way that its motion becomes periodic. The state of the rotors of the two Figs. are the same, except on the top row and the right column, but these sites are not visited by the particle as shown in the next Fig. Ra´ ul Rechtman (IER-UNAM) Deterministic walks 16 / 53
Initially ordered flipping rotor landscape 51 49 47 45 43 41 20 22 24 26 Trajectory of the walker between t = T 0 and t = T 1 to be compared with the previous Figs. At t = T 0 the walker is in (25 , 50), the upper right red circle, and at t = T 0 + T 1 , the walker is in (23 , 48), the lower left circle. Ra´ ul Rechtman (IER-UNAM) Deterministic walks 17 / 53
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